No. Audacity will accept multi-track mixer/interfaces (4-track) for recording, but you don't have a multi-track mixer—that we can tell.

If you're all in the same non-studio room, you will have zero chance to isolate each voice from the others anyway. So your only chance to pull this off is manually mix the show on good headphones while it's going. If you've never done that before, it will be a challenge.

If you want to beat this up more, we'll need the mixer maker and model number and who made the microphones. You're not in a studio or quiet room, are you?

Also remember the forum elves are sprayed across nine time zones and not all of us are into live recording.

If I recall correctly, the BEHRINGER XENYX X1204USB can only record 2 audio channels (stereo).Audacity can only record from one device at a time, so with the equipment you have described, you will only be able to record a maximum of 2 independent audio channels.

To record more the 2 channels, you would need to use a sound card with more input channels. Unfortunately we don't know which multi-channel sound cards work with Audacity - some don't because they require ASIO for multi-channel recording, and Audacity does not support ASIO.

Another option would be to record on a hardware multi-channel recorder (example: Boss BR-800, or Zoom R-16).

That doesn't mean you can't force the mixer into giving you three independent voices. You can. This mixer has two "sends." The red and orange knobs.

So. Leave them down on microphones one and two. Fade those normally so one is Left and the other Right. Feed those out to the computer for recording normally. For Microphone three, you leave the slide fader down, but turn up, say FX Send (the orange knob). FX Send has a master on the right of the console. Turn that up, too. Plug the second recorder into AUX Send 2-FX on the top row of connectors.

It will be a juggling act to start and stop the recorders and as near as I can tell, FX Send has no volume meters, so you'll need to use the ones on the second recorder.

Are you following all this?

Sound Sync is going to be a really big deal, so just after you start both recorders and before they start performing, yell "SOUND SYNC! and clap your hands once. That sets the sync mark much like the movie people do.

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Now you know what the movie clap board does.

It will be insanely handy to do that at the end, too, but nobody is going to remember to do that.

Yes it does work. I've done that on a complicated shoot.

You can record Mic 3 with with a second computer by adding a behringer UCA-202 or equivalent. That's an analog to USB converter. Plugging directly into the computer MIC-IN is not recommended.

I don't mean to suggest you're out of the woods even if you get all that to work. Each microphone is going to capture one performer's good voice and the bare-wall echo of the other two. Any correction you make to one voice is going to affect the echo of the others.

And if you do use the second computer technique for the third microphone, sound sync is super important because if you don't get that right, the show will have two sets of "wrong voice" echoes, not one and it may get worse as the show progresses.